How do you convert 3.142 (142 repeating) as a fraction?

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Answer 1 · 2016-03-10T22:38:25

$3 . ¯ ¯¯¯¯ ¯ 142 = \frac{3139}{999}$ $3.bar(142) = 3139/999$

Multiply by $1000 - 1$ $1000-1$ to shift the number three places to the left and subtract the original to eliminate the repeating decimal tail...

$(1000 - 1) \cdot 3 . ¯ ¯¯¯¯ ¯ 142 = 3142 . ¯ ¯¯¯¯ ¯ 142 - 3 . ¯ ¯¯¯¯ ¯ 142 = 3139$ $(1000-1)*3.bar(142) = 3142.bar(142) - 3.bar(142) = 3139$

Dividing both sides by $(1000 - 1)$ $(1000-1)$ we find:

$3 . ¯ ¯¯¯¯ ¯ 142 = \frac{3139}{1000 - 1} = \frac{3139}{999}$ $3.bar(142) = 3139 / (1000-1) = 3139/999$

This improper fraction is in lowest terms, since:

$3139 = 43 \cdot 73$ $3139 = 43 * 73$

$999 = 3 \cdot 3 \cdot 3 \cdot 37$ $999 = 3*3*3*37$

So $3139$ $3139$ and $999$ $999$ have no common factors larger than $1$ $1$ .